Introduction to Piecewise Differentiable Equations [electronic resource] /
Scholtes, Stefan.author.
This brief provides an elementary introduction to the theory of piecewise differentiable functions with an emphasis on differentiable equations.  In the first chapter, two sample problems are used to motivate the study of this theory. The presentation is then developed using two basic tools for the analysis of piecewise differentiable functions: the Bouligand derivative as the nonsmooth analogue of the classical derivative concept and the theory of piecewise affine functions as the combinatorial tool for the study of this approximation function. In the end, the results are combined to develop inverse and implicit function theorems for piecewise differentiable equations.  This Introduction to Piecewise Differentiable Equations will serve graduate students and researchers alike. The reader is assumed to be familiar with basic mathematical analysis and to have some familiarity with polyhedral theory.
Description based upon print version of record.
Introduction to Piecewise Differentiable Equations; Preface; Contents; Chapter 1 Sample Problems for Nonsmooth Equations; 1.1 Complementarity Problems; 1.1.1 Equilibria of Dynamical Systems; 1.1.2 Nonlinear Complementarity Problems; 1.1.3 Comments and References; 1.2 Stationary Solutions of Parametric Programs; 1.2.1 Multiobjective Optimization; 1.2.2 The Kojima Mapping; 1.2.3 Comments and References; 1.3 Appendix: Differentiability Versus Nondifferentiability; Chapter 2 Piecewise Affine Functions; 2.1 Elements from Polyhedral Theory; 2.1.1 Convex Sets and Convex Cones
2.1.2 Polyhedral Sets and Polyhedral Cones2.1.3 The Face Lattice of a Polyhedron; 2.1.4 Comments and References; 2.2 Basic Notions and Properties; 2.2.1 Representations of Piecewise Affine Functions; 2.2.2 Local Approximations of Piecewise Affine Functions; 2.2.3 Lipschitz Continuity of Piecewise Affine Functions; 2.2.4 Comments and References; 2.3 Piecewise Affine Homeomorphisms; 2.3.1 Coherently Oriented Piecewise Affine Functions; 2.3.2 Piecewise Affine Local Homeomorphisms; 2.3.3 The Factorization Lemma; 2.3.4 The Branching Number Theorem; 2.3.5 Comments and References
2.4 Euclidean Projections2.4.1 The Euclidean Projection onto a Polyhedron; 2.4.2 The Normal Manifold; 2.4.3 An Application: Affine Variational Inequalities; 2.4.4 Comments and References; 2.5 Appendix: The Recession Function; Chapter 3 Elements from Nonsmooth Analysis; 3.1 The Bouligand Derivative; 3.1.1 The B-Derivative of a Locally Lipschitz Continuous Function; 3.1.2 Strongly B-Differentiable Functions; 3.1.3 Comments and References; 3.2 Inverse and Implicit Function Theorems; 3.2.1 B-Derivatives of Local Lipschitz Homeomorphisms; 3.2.2 An Inverse Function Theorem; 3.2.3 Hadamard's Theorem
3.2.4 Comments and References3.3 Appendix: Inverse Function Theorems of Clarke and Kummer; Chapter 4 Piecewise Differentiable Functions; 4.1 Basic Notions and Properties; 4.1.1 Local Lipschitz Continuity; 4.1.2 B-Differentiability; 4.1.3 Strong B-Differentiability; 4.1.4 Continuous Differentiability; 4.1.5 Comments and References; 4.2 Piecewise Differentiable Homeomorphisms; 4.2.1 An Implicit Function Theorem; 4.2.2 A Bound for the Condition Number; 4.2.3 Comments and References; 4.3 Appendix: A Formula for the Generalized Jacobian; Chapter 5 Sample Applications
5.1 Variational Inequalities and Normal Maps5.1.1 A Homeomorphism Condition for Normal Maps of Polyhedra; 5.1.2 Comments and References; 5.2 Sensitivity Analysis for Mathematical Programs; 5.2.1 Sensitivity Analysis of Stationary Solutions; 5.2.2 Sensitivity Analysis of Local Minimizers; 5.2.3 Comments and References; References; Index
Includes bibliographical references and index.
This brief provides an elementary introduction to the theory of piecewise differentiable functions with an emphasis on differentiable equations.  In the first chapter, two sample problems are used to motivate the study of this theory. The presentation is then developed using two basic tools for the analysis of piecewise differentiable functions: the Bouligand derivative as the nonsmooth analogue of the classical derivative concept and the theory of piecewise affine functions as the combinatorial tool for the study of this approximation function. In the end, the results are combined to develop inverse and implicit function theorems for piecewise differentiable equations.  This Introduction to Piecewise Differentiable Equations will serve graduate students and researchers alike. The reader is assumed to be familiar with basic mathematical analysis and to have some familiarity with polyhedral theory.
Mathematical analysis.
Analysis (Mathematics).
Functions of complex variables.
Calculus of variations.
Analysis.
Functions of a Complex Variable.
Calculus of Variations and Optimal Control; Optimization.